2-microlocal analysis of martingales and stochastic integrals

Recently, a new approach in the fine analysis of sample paths of stochastic processes has been developed to predict the evolution of the local regularity under (pseudo-)differential operators. In this paper, we study the sample paths of continuous martingales and stochastic integrals. We proved that the almost sure 2-microlocal frontier of a martingale can be obtained through the local regularity of its quadratic variation. It allows to link the Hölder regularity of a stochastic integral to the regularity of the integrand and integrator processes. These results provide a methodology to predict the local regularity of diffusions from the fine analysis of its coefficients. We illustrate our work with examples of martingales with unusual complex regularity behaviour and square of Bessel processes.     

Regularity of Solutions to Bilinear Stochastic Wave Equations

Abstract

The paper is concerned with strong solutions of bilinear stochastic wave equations in ? ^d , of which the coefficients contain semimartingale white noises with spatial parameters. For the Cauchy problems, the existence and spatial regularity of solutions in Sobolev spaces are proved under appropriate conditions. The dependence of solution regularity on the smoothness of the random coefficients is ascertained. The proofs are based on stochastic energy inequalities, the semigroup method and certain submartingale inequalities. Regularity results are also obtained for the special case of Wiener semimartingales.     

On the Besov regularity of periodic Lévy noises

In this paper, we study the Besov regularity of Lévy white noises on the d-dimensional torus. Due to their rough sample paths, the white noises that we consider are defined as generalized stochastic fields. We, initially, obtain regularity results for general Lévy white noises. Then, we focus on two subclasses of noises: compound Poisson and symmetric-α-stable (including Gaussian), for which we make more precise statements. Before measuring regularity, we show that the question is well-posed; we prove that Besov spaces are in the cylindrical σ-field of the space of generalized functions. These results pave the way to the characterization of the n-term wavelet approximation properties of stochastic processes.     

Regularity of Backward Stochastic Volterra Integral Equations in Hilbert Spaces

This article investigates backward stochastic Volterra integral equations in Hilbert spaces. The existence and uniqueness of their adapted solutions is reviewed. We establish the regularity of the adapted solutions to such equations by means of Malliavin calculus. For an application, we study an optimal control problem for a stochastic Volterra integral equation driven by a Hilbert space-valued fractional Brownian motion. A Pontryagin-type maximum principle is formulated for the problem and an example is presented.     

Metric regularity of composition set-valued mappings: Metric setting and coderivative conditions

The paper concerns a new method to obtain a proof of the openness at linear rate/metric regularity of composite set-valued maps on metric spaces by the unification and refinement of several methods developed somehow separately in several works of the authors. In fact, this work is a synthesis and a precise specialization to a general situation of some techniques explored in the last years in the literature. In turn, these techniques are based on several important concepts (like error bounds, lower semicontinuous envelope of a set-valued map, local composition stability of multifunctions) and allow us to obtain two new proofs of a recent result having deep roots in the topic of regularity of mappings. Moreover, we make clear the idea that it is possible to use (co)derivative conditions as tools of proof for openness results in very general situations.     

A remark on regularity of 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity

In this paper, we consider one-dimensional compressible isentropic Navier-Stokes equations with the viscosity depending on density and with the free boundary. The viscosity coefficient μ is proportional to ρθ with θ>0, where ρ is the density. The existence, uniqueness, regularity of global weak solutions in H¹([0,1]) have been established by Xin and Yao in [Z. Xin, Z. Yao, The existence, uniqueness and regularity for one-dimensional compressible Navier-Stokes equations, preprint]. Furthermore, under certain assumptions imposed on the initial data, we improve the regularity result obtained in [Z. Xin, Z. Yao, The existence, uniqueness and regularity for one-dimensional compressible Navier-Stokes equations, preprint] by driving some new a priori estimates.     

A remark on the regularity criterion of Boussinesq equations with zero heat conductivity

In this note, we consider the regularity problem under the critical condition to the Boussinesq equations with zero heat conductivity. The Serrin type regularity criteria are established in terms of the critical Besov spaces. This improves a result established in a recent work by Geng and Fan (2012)  [6].     

Metric regularity and systems of generalized equations

The paper is devoted to a revision of the metric regularity property for mappings between metric or Banach spaces. Some new concepts are introduced: uniform metric regularity and metric multi-regularity for mappings into product spaces, when each component is perturbed independently. Regularity criteria are established based on a nonlocal version of Lyusternik-Graves theorem due to Milyutin. The criteria are applied to systems of generalized equations producing some “error bound” type estimates.     

Remarks on the regularity criteria for generalized MHD equations

We study the Cauchy problem for the generalized MHD equations, and prove some regularity criteria involving the integrability of ∇u in the Morrey, multiplier spaces.     

A Generalized Blow up Criteria with One Component of Velocity for 3D Incompressible MHD System∗

In this paper, the authors study the global regularity of the 3D magnetohydrodynamics system in terms of one velocity component. In particular, they establish a new Prodi-Serrin type regularity criterion in the framework of weak Lebesgue spaces both in time and space variables.     

New characterizations of Besov and Triebel-Lizorkin spaces over spaces of homogeneous type

Using the T1 theorem for the Besov and Triebel-Lizorkin spaces, we give new characterizations of Besov and Triebel-Lizorkin spaces with minimum regularity and cancellation conditions over spaces of homogeneous type.     

Geometric constrains for global regularity of 2D quasi-geostrophic flows

We study the two-dimensional quasi-geostrophic equations (2D QG) in Sobolev spaces. We first prove a new analytic condition for global regularity which is both sufficient and necessary. We then prove several new results on the geometric constraints on the 2D QG active scalar which suppress the development of singularity from the nonlinear stretching mechanism. We focus mainly on the case with critical dissipation. Our results are also relevant to the inviscid case.     

On Regularity of Incompressible Fluid with Shear Dependent Viscosity

The authors consider a non-Newtonian fluid governed by equations with p-structure in a cubic domain.A fluid is said to be shear thinning(or pseudo-plastic) if 1 < p < 2,and shear thickening(or dilatant) if p > 2.The case p > 2 is considered in this paper.To improve the regularity results obtained by Crispo,it is shown that the secondorder derivatives of the velocity and the first-order derivative of the pressure belong to suitable spaces,by appealing to anisotropic Sobolev embeddings.     

On the global regularity of shear thinning flows in smooth domains

In some recent papers we have been pursuing regularity results up to the boundary, in W2,l(Ω) spaces for the velocity, and in W1,l(Ω) spaces for the pressure, for fluid flows with shear dependent viscosity. To fix ideas, we assume the classical non-slip boundary condition. From the mathematical point of view it is appropriate to distinguish between the shear thickening case, p>2, and the shear thinning case, p<2, and between flat-boundaries and smooth, arbitrary, boundaries. The p<2 non-flat boundary case is still open. The aim of this work is to extend to smooth boundaries the results proved in reference [H. Beirão da Veiga, On non-Newtonian p-fluids. The pseudo-plastic case, J. Math. Anal. Appl. 344 (1) (2008) 175-185]. This is done here by appealing to a quite general method, introduced in reference [H. Beirão da Veiga, On the Ladyzhenskaya-Smagorinsky turbulence model of the Navier-Stokes equations in smooth domains. The regularity problem, J. Eur. Math. Soc., in press], suitable for considering non-flat boundaries.     

A Quantitative Regularity Estimate for Nonnegative Supersolutions of Uniformly Parabolic Equations

This note establishes an interior quantitative lower bound for nonnegative supersolutions of fully nonlinear uniformly parabolic equations. The result may be interpreted as a quantitative version of a growth lemma established by Krylov and Safonov for nonnegative supersolutions of linear uniformly parabolic equations in nondivergence form. Our approach is different, and follows from an application of a reverse Holder inequality. The result is the parabolic analogue of an elliptic regularity estimate established by Caffarelli, Souganidis, and Wang in the stochastic homogenization of fully nonlinear uniformly elliptic equations.     

Optimal regularity estimates for parabolic p-harmonic equations

In this paper optimal regularity estimates for weak solutions of quasilinear parabolic equations of p-Laplacian type with small BMO coefficients are investigated. Our results improve the known results for such equations using a harmonic analysis free technique.     

A note on maximal estimates for stochastic convolutions

In stochastic partial differential equations it is important to have pathwise regularity properties of stochastic convolutions. In this note we present a new sufficient condition for the pathwise continuity of stochastic convolutions in Banach spaces.     

Stochastic evolution equations in Hilbert spaces

In this paper we consider Ito's stochastic differential equation in Hilbert spaces. A strong solution is generated by the difference approximation. A regularity result is obtained for solutions to a class of parabolic stochastic partial differential equations. Hyperbolic stochastic evolution equations are also discussed.     

A remark on regularity criterion for the dissipative quasi-geostrophic equations

This paper concerns with a regularity criterion of solutions to the 2D dissipative quasi-geostrophic equations. Based on a logarithmic Sobolev inequality in Besov spaces, the absence of singularities of θ in [0,T] is derived for θ a solution on the interval [0,T) satisfying the condition

     

A regularity criterion for the 2D MHD system with zero magnetic diffusivity

This note proves a regularity criterion ∇b∈L¹(0,T;BMO(R²)) for the 2D MHD system with zero magnetic diffusivity.